The geometries � of
soccer balls
David Swart�April 11, 2019 dmswart.com
< 1970s
> 1970s
1974
≥ 1970s
1980s
1980s
Icosahedron
Truncated icosahedron
Spherical truncated icosahedron
C60 Buckminsterfullerine “Bucky ball”
Goldberg Polyhedra
(1, 0)
(1, 1)
(7, 0)
(2, 0)
Platonic Solids
Tetrahedron
Octahedron
Cube
Icosahedron
Dodecahedron
Archimedean Solids
Non-Euclidean surfaces
hyperbolic soccer ball image by Keith Henderson
1970s - 1990s
photos: soccerballworld.com
> 2000s
Archimedean Collection
Archimedean Collection
2 Rosette Groups
7 Frieze Groups
images by Andrew Glasner
17 Wallpaper Groups
Images: The Grammar of Ornament�Owen Jones
14 Spherical Symmetry Groups
Polyhedral Symmetry
Lines of Reflection
Kaleidoscope points
Orbifold:
*532
Centers of rotation
Centers of rotation
Orbifold:
432
Glide reflections
Orbifold:
❌
*532
*532
*432
*432
*332
532
532
432
432
332
332
3*2
3*2
3*2 (Pyritohedral)
3*2
Rob Lavinsky, iRocks.com
3*2
3*2
Polyhedral Symmetry
Cyclic and Dihedral Symmetry
Spherical Frieze Patterns
Cyclic and Dihedral Symmetry
*N22
*N22
2*N
2*N
N22
N22
N22
N22
222
422
522
NN
NN
N*
N*
*NN
*NN
N❌
Spherical Frieze Patterns
Spherical Symmetry Groups
N❌
N×
14 Soccerball Groups
14 Soccerball Groups
Degenerate cases
1
O(3)
C ∞
D∞
Honorable mentions
How to
draw a
soccer ball
#1 ✓Look at one
#1 ✓Look at one
#2 ✗Don’t have a mishmash of polygons
#3 ✗ Don’t fill it with all hexagons
#3 ✗ Don’t fill it with all hexagons
#4 ✓Color the pentagons black
#5 ✓Pentagon to pentagon bridge
Counterexamples
18 Octagons (12 are concave)
Two hexagons with six 180° corners
Deltoidal Icositetrahedron
Spherical Deltoidal Icositetrahedron
Spherical Tessellation
An all hexagon soccer ball design
The geometries � of
soccer balls
David Swart�April 11, 2019 dmswart.com